A note on embeddings of 3-manifolds in symplectic 4-manifolds

TL;DR

This paper demonstrates that all closed oriented 3-manifolds can be embedded in simply-connected symplectic 4-manifolds, with some obstructions to smooth embeddings, and explores implications for homology cobordism and exotic smooth structures.

Contribution

It proves topological embeddings of 3-manifolds into symplectic 4-manifolds and identifies obstructions to smooth embeddings, advancing understanding of 3- and 4-manifold interactions.

Findings

Any closed oriented 3-manifold can be topologically embedded in a simply-connected symplectic 4-manifold.

After stabilization, the embedding can be made smooth.

Obstructions exist for smooth embeddings respecting orientations in certain symplectic 4-manifolds.

Abstract

We show that any closed oriented 3-manifold can be topologically embedded in some simply-connected closed symplectic 4-manifold, and that it can be made a smooth embedding after one stabilization. As a corollary of the proof we show that the homology cobordism group is generated by Stein fillable 3-manifolds. We also find obstructions on smooth embeddings: there exists 3-manifolds which cannot smoothly embed in a way that appropriately respect orientations in any symplectic manifold with weakly convex boundary. This embedding obstruction can also be used to detect exotic smooth structures on 4-manifolds.